So this is a really interesting question, but one that can't really be answered definitively, given that actually tracing your family's history is beyond the scope of the subreddit, not least because it would involve visiting any number of (shut) physical archives. However, I can provide some context and speculation.

You're certainly right that the most likely source of the badge reaching Spain is the civil war of 1936-9. The war saw significant international involvement and intervention, including by a relatively small contingent from the Soviet Union (just over 2000 all told, up to about 800 in Spain at any one time). They were part of the ongoing Soviet effort to support the Spanish Republican government against the military uprising launched in July 1936. Catalonia, as you'll no doubt be aware, was firmly in the Republican camp, falling only in the final months of the war, so there were certainly Soviet personnel working in the region.

The Soviet military mission is the first potential source for the badge. Most of its members were, naturally, military specialists - pilots and tank drivers being the most common. However, there were significant numbers of engineers and other technical workers, a smaller number of encryption specialists and a very small number of chemists (generally fuel specialists, as far as I can tell). While many of these were employed in military roles, some also worked in industry, advising on the conversion of civilian factories to military uses, for instance. As such, it is far from impossible that your ancestor came into contact either with one of these specialists or someone who worked with them (the latter, of course, being far from unlikely to have ended up in a Francoist prison, though it seems less likely that they would have kept incriminating material on them if so). Trading little mementos and keepsakes would hardly be unlikely in either context though.

The other likely source is through Soviet cultural outreach. As part of its intervention, the USSR launched a large publicity (or propaganda) campaign in Republican Spain, seeking to build on the goodwill and political capital they had gained through supporting the Republic militarily. This involved exporting (and translating) Soviet media, publications, art, literature and other forms of cultural exchange. This included science: news of Soviet scientific advances were given prominent place within these efforts, and certainly included sending copies of scientific journals, and possibly other forms of scientific ephemera. This was done in tandem to the military intervention, generally run through the Soviet embassy in coordination with local friendship societies. Though these efforts generally petered out by 1938, they did happen across the Republican Zone, including Catalonia (though the USSR didn't really have anyone who could speak Catalan who could translate the material, which proved a handicap - in fairness, they also had this problem with Castilian, and a lot of the propaganda material arrived instead in a variety of Western European languages such as French, English or Italian, or occasionally simply not translated at all - like this badge, for that matter).